# Periodic orbits of Hamiltonian systems

Consider a second order Hamiltonian system of the type $$\ddot{x}+V'(x)=0, \quad x \in \mathbb{R}^N.$$ Under very `natural assumptions' it is possible to prove the existence of a non constant $T$-periodic orbit $\varphi^T=(\varphi_{1}^{T},\ldots,\varphi_{N}^{T})$ for any given $T>0$. If $N \ge 2$ it may as well happen that $\varphi_{i}^{T} \equiv \text{ constant }$ for some index $i$.

Is there any result that guarantees the existence of a $\varphi^T$ for which all the components are non constant functions?

• Maybe I'm misunderstanding, but I don't see how you get a nonconstant periodic orbit for an arbitrary period. For instance if you set $V'(x)=x$ then all the periodic orbits seem to have period $2\pi$. More generally, assuming that $V'$ is Lipschitz, the Yorke estimate (applied to the corresponding first-order system on $R^{2N}$) gives an explicit positive lower bound for the minimal period of a periodic orbit. Am I misinterpreting what you wrote? Or maybe your "natural assumptions" involved $V'$ growing faster than linearly, though this would seem a bit unphysical... Commented Jan 19, 2012 at 3:02
• I'm actually refering to those assumptions made by Rabinowitz to prove what I just claimed! Commented Jan 19, 2012 at 9:06
• so, you want conditions on $V$ that ensure the existence of a T-periodic orbit with non-constant coordinates for any prescribed period T, or would it be OK the existence of just a periodic orbit with non-constant coordinates? Commented Jan 19, 2012 at 9:21
• Thanks for the precision. I want conditions for the existence of a T-periodic orbit having a prescribed period and whose coordinates are non constants Commented Jan 19, 2012 at 9:49